For a connected graph $\mathcal {G}$ , the distance $d(u, v)$ between any two vertices $u$ , $v~\in ~\mathcal {V(G)}$ is defined as; $d(u,v)=\min _{\mathcal {P}_{u,v}}\{\text {length of}~\mathcal {P}_{uv}\}$ i.e the minimum length of path connecting any two vertices. Two vertices $u$ , $v$ of a graph $G$ are said to be resolved by a vertices $\mathcal {W}$ of a graph $\mathcal {G}$ if $d(u,w) \neq d(v,w)$ . An ordered set $\mathcal {W} = \{w_{1},w_{2},w_{3},\ldots,w_{k}\}\,\,\in \,\,V(G)$ is resolving set for a connected graph $G$ , if for any two vertices $u$ , $v$ there exists $w_{i} \in W$ such that $d(u,w_{i}) \neq d(v, w_{i})$ . The representation of vertices $v$ w.r.t $\mathcal {W}$ is represented by $r(v|\mathcal {W})$ which is $k$ -vector( $k$ -tuple) $(d(v,w_{1}),d(v,w_{2}),d(v,w_{3}),\ldots,d(v,w_{k}))$ . If the representation of all the vertices of graph $\mathcal {G}$ are different w.r.t $\mathcal {W}$ then $\mathcal {W}$ is the resolving set for $G$ [1] . A resolving set that contains minimum numbers of vertices is metric basis for $\mathcal {G}$ . The cardinality of smallest resolving set is the metric dimension of $\mathcal {G}$ , represented by $dim(\mathcal {G})$ [2] , [3] . A family of connected graph $\mathcal {G}$ has unbounded metric dimension if $dim(\mathcal {G})$ depends on order of a graph. In this paper, we show that total graph of path power three and four have unbounded metric dimension. We also proved some results on edges of power of path graph.

We propose a new fractional-order (space and time) total variation regularized model for multiplicative noise removal in this research article. We use the regularly varying fuzzy membership degrees to characterize the likelihood of a pixel related to edges, texture regions, and flat regions to improve model efficiency. This approach is capable of maintaining edges, textures, and other image information while significantly reducing the blocky effect. We opt for the option of local actions. In order to efficiently find the minimizer of the prescribed energy function, the semi-implicit gradient descent approach is used (which derives the corresponding fractional-order Euler-Lagrange equations). The existence and uniqueness of a solution to the suggested variational model are proved. Experimental results show the efficiency of the suggested model in visual enhancement, preserving details and reducing the blocky effect while extracting noise as well as an increase in the PSNR (dB), SSIM, relative error, and less CPU time(s) comparing to other schemes.

This article introduces a fast meshless numerical algorithm to solve nonlinear partial di?erential equations (PDE) through Radial Basis Functions (RBFs) approximation connected with the Total Variation (TV)-based minimization functional and to show its application to image denoising containing multiplicative noise. These features in the proposed algorithm not only to remove the noise from the images and preserve the edges but also to reduce the staircase effect extensively from images that cause the image to look blocky. It is worth mentioning that the proposed approach can be easily employed for nonlinear problems due to the lack of dependence on a mesh or integration procedure. The numerical experiments and comparative examples illustrate the effectiveness of the proposed algorithm regarding the robustness and visual improvement as well as peak-signal-to-noise ratio (PSNR), signal-to-noise ratio (SNR), and mean structural similarity index (MSSIM) compared to the recent traditional TV-based methods.

This paper explores the domination number of the splitting graph associated with higher powers of cycle graphs. We begin by computing the domination number of the splitting graph of the square and cube of the cycle graph C_ξ. Through analysis and observed patterns, we propose a general conjecture regarding the domination number of the splitting graph of the κ^th power ofC_ξ. These results extend the existing work on domination parameters in transformed graphs and provide a foundation for further theoretical development and applications.

For a simple connected graph G, the distance d(ai, bi), where ai, bi ∈ V (G), is the length of the shortest path, measured by the number of edges, between vertices ai and bi. The n-order partition of vertices of G is denoted as ψ = {ψ1, ψ2, ψ3, . . . , ψn}. Thenotation of vertex ai ∈ G with respect to ψ is the vector code {d(ai, ψ1), d(ai, ψ2), . . . , d(ai, ψn)}.Partition set ψ is called a resolving partition set if the representation of each vertex with respect to ψ is unique. The partition dimension of G is defined as the minimum size of such a resolving partition set. In this research, we investigated the partition di- mension of the generalized gear graph G(k, n) and the generalized fan graph F (2, n)..